Find the value of sin ^{-1}left(sin frac{2 pi | vedclass

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Question

(C) We need to evaluate $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$.The principal value branch of $\sin ^{-1} x$ is $\left[-\frac{\pi}{2}, \frac{\pi

Vedclass · Accepted Answer

$\frac{\pi}{3}$

Vedclass · Answer

(C) We need to evaluate $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$.The principal value branch of $\sin ^{-1} x$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.Since $\frac{2 \pi}{3} \approx 120^\circ$,which is not in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,we use the property $\sin(\pi - x) = \sin x$.Therefore,$\sin \frac{2 \pi}{3} = \sin \left(\pi - \frac{2 \pi}{3}\right) = \sin \frac{\pi}{3}$.Now,$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right) = \sin ^{-1}\left(\sin \frac{\pi}{3}\right)$.Since $\frac{\pi}{3} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,we have $\sin ^{-1}\left(\sin \frac{\pi}{3}\right) = \frac{\pi}{3}$.Thus,the correct value is $\frac{\pi}{3}$.

Find the value of $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$.

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